To determine which polynomial is in standard form, we need to check whether the terms of the polynomial are arranged in descending order of their powers of [tex]\( x \)[/tex].
Polynomials given:
1. [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]
2. [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]
3. [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]
4. [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
Let's analyze each polynomial to see if the powers of [tex]\( x \)[/tex] are arranged from highest to lowest:
1. [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]:
- Contains terms: [tex]\( 6x^3, -8x^2, 2x, 1 \)[/tex]
- Ordered form: [tex]\( 6x^3 - 8x^2 + 2x + 1 \)[/tex]
- This is not in standard form.
2. [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]:
- Contains terms: [tex]\( 6x^3, 2x^2, -9x, 12 \)[/tex]
- Ordered form: [tex]\( 6x^3 + 2x^2 - 9x + 12 \)[/tex]
- This is not in standard form.
3. [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]:
- Contains terms: [tex]\( 6x^3, -3x^2, 5x, 2 \)[/tex]
- Ordered form: [tex]\( 6x^3 - 3x^2 + 5x + 2 \)[/tex]
- This is not in standard form.
4. [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]:
- Contains terms: [tex]\( 2x^3, 4x^2, -7x, 5 \)[/tex]
- Ordered form: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
- This is in standard form.
After analyzing all the polynomials, we conclude that none of them were originally written in standard form. The indices of the polynomials that are in standard form yield an empty list [tex]\([]\)[/tex]. Therefore, no polynomial in the given list is in standard form as provided in the options.
